Theorem nnsplit

Description: Express the set of positive integers as the disjoint (see nnuzdisj ) union of the first N values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Assertion	nnsplit	⊢ ( 𝑁 ∈ ℕ → ℕ = ( ( 1 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
2	1	a1i	⊢ ( 𝑁 ∈ ℕ → ℕ = ( ℤ≥ ‘ 1 ) )
3		peano2nn	⊢ ( 𝑁 ∈ ℕ → ( 𝑁 + 1 ) ∈ ℕ )
4	3 1	eleqtrdi	⊢ ( 𝑁 ∈ ℕ → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 1 ) )
5		uzsplit	⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ℤ≥ ‘ 1 ) = ( ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) )
6	4 5	syl	⊢ ( 𝑁 ∈ ℕ → ( ℤ≥ ‘ 1 ) = ( ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) )
7		nncn	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ )
8		1cnd	⊢ ( 𝑁 ∈ ℕ → 1 ∈ ℂ )
9	7 8	pncand	⊢ ( 𝑁 ∈ ℕ → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 )
10	9	oveq2d	⊢ ( 𝑁 ∈ ℕ → ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) = ( 1 ... 𝑁 ) )
11	10	uneq1d	⊢ ( 𝑁 ∈ ℕ → ( ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ( ( 1 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) )
12	2 6 11	3eqtrd	⊢ ( 𝑁 ∈ ℕ → ℕ = ( ( 1 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) )